Our torus starts out with a purely toroidal magnetic field. Shear causes the field to go unstable to the toroidal MRI, which leads to magnetized turbulence and viscous stresses, driving accretion of gas on to the BH. Figure REF (a) shows that the gas reaches the BH at \(t \sim 600 r_g/c\) , and mass accretion rate saturates at \(t \gtrsim 3,\!000r_g/c\)  and remains approximately constant until \(t\sim 10^4r_g/c\) . Figure REF (b) shows that at this time the only outflow present is a sub-relativistic wind with
energy efficiency \(\eta _{\rm wind} \approx 5\%\) .
Figure REF (c) shows that the poloidal magnetic flux on
the BH, \(\Phi _{BH} = 0.5\int _{r=r_{\rm H}}\left|B^r\right| dA_{\theta \phi }\) , grows from 0 to 20, as shown by the blue line. Here, the integral is over both hemispheres of the event horizon, \(r_{\rm H}=r_g[1+(1-a^2)^{1/2}]\) , and the factor of \(0.5\)  converts it to one hemisphere [1]}.
The reservoir of positive poloidal magnetic flux in the disc, \(\Phi _{\rm disc}=\max _{r} \Phi _{p}(r)\)  with \(\Phi _{p}(r)=\max _{\theta }\int _0^\theta B^r dA_{\theta \phi }\) , shown with the red line, also keeps growing, pointing to a large-scale dynamo activity in the disc.
Figure REF (a) shows a time sequence illustrating the generation of poloidal magnetic flux loops by the MHD turbulence (see the movie in the SI): several loops form just outside the equatorial plane, grow in strength, and buoyantly rise away from the equator. This process is stochastic: one of the loops ends up taking over the inner \(100r_g\)  of the disc with the others getting expelled. This is consistent with the \(\alpha {-}\Omega \)  large-scale poloidal magnetic flux dynamo [2]}, [3]}, in which a toroidal magnetic field loop undergoes Parker instability, buoyantly rises, and the Coriolis force twists it into a poloidal magnetic field loop. This way, the \(\alpha \) -effect can convert toroidal into poloidal flux. The \(\Omega \) -effect then does the opposite, shearing out this freshly-generated poloidal magnetic flux loop into toroidal magnetic flux, and thereby completing the positive feedback cycle. This is a possible mechanism for both the initial formation of the poloidal magnetic flux loops and their subsequent runaway growth in strength and size, as seen in our simulations.
